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Backpropagation-free Training of Deep Physical Neural Networks

Ali Momeni, Babak Rahmani, Matthieu Mallejac, Philipp Del Hougne, Romain Fleury

arXiv:2304.11042v3cs.LGcs.NEphysics.app-phphysics.optics

TL;DR

Backpropagation is unsuitable for physical neural networks because its physical implementations are complex and difficult to scale. The paper proposes a simple physics-compatible architecture with biologically plausible, backpropagation-free training, reporting speed advantages while noting that challenges remain before physical networks can replace digital counterparts.

  • Problem

    Backpropagation is unsuitable for physical neural networks because its physical implementations are complex and lack scalability.

  • Method

    The paper proposes a simple, physics-compatible physical neural-network architecture augmented by biologically plausible backpropagation-free training.

  • Results

    The method produces a significant speed-up during both inference and training and outperforms in-silico training and the state-of-the-art PA-BP method.

  • Takeaways & Limitations

    The work demonstrates backpropagation-free training of physical neural networks while addressing challenges associated with physical implementations of backpropagation.

  • Takeaways & Limitations

    Challenges remain before physical neural networks can replace their digital counterparts, and how this replacement will occur is not yet clear.

Abstract

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Recent years have witnessed the outstanding success of deep learning in various fields such as vision and natural language processing. This success is largely indebted to the massive size of deep learning models that is expected to increase unceasingly. This growth of the deep learning models is accompanied by issues related to their considerable energy consumption, both during the training and inference phases, as well as their scalability. Although a number of work based on unconventional physical systems have been proposed which addresses the issue of energy efficiency in the inference phase, efficient training of deep learning models has remained unaddressed. So far, training of digital deep learning models mainly relies on backpropagation, which is not suitable for physical implementation as it requires perfect knowledge of the computation performed in the so-called forward pass of the neural network. Here, we tackle this issue by proposing a simple deep neural network architecture augmented by a biologically plausible learning algorithm, referred to as "model-free forward-forward training". The proposed architecture enables training deep physical neural networks consisting of layers of physical nonlinear systems, without requiring detailed knowledge of the nonlinear physical layers' properties. We show that our method outperforms state-of-the-art hardware-aware training methods by improving training speed, decreasing digital computations, and reducing power consumption in physical systems. We demonstrate the adaptability of the proposed method, even in systems exposed to dynamic or unpredictable external perturbations. To showcase the universality of our approach, we train diverse wave-based physical neural networks that vary in the underlying wave phenomenon and the type of non-linearity they use, to perform vowel and image classification tasks experimentally.

Introduction

Physical neural networks offer scalable, energy-efficient hardware by using physical transformations, but their training remains difficult because backpropagation requires complex, scalable hardware implementations and complete knowledge of the forward computation graph. The paper introduces model-free forward-forward training to directly train deep physical networks without detailed models of their nonlinear layers.

  • Motivation: Physical neural networks can improve scalability, energy efficiency, and speed by leveraging physical transformations instead of conventional software–hardware divisions.Increasing physical-network depth may also expand network expression capabilities.
  • Limitations of existing training: Backpropagation is unsuitable for many physical networks because implementing its operations in hardware is complex and difficult to scale.Some implementations are restricted to lossless and reciprocal systems, while others require multiple propagation rounds that many setups cannot achieve.
  • Limitations of existing training: Physics-aware backpropagation reduces some in-silico limitations but still requires a differentiable digital backward model, causing slower training, higher power use, and extra memory requirements.Strong physical perturbations may also make reusing or fine-tuning trained models difficult, sometimes requiring retraining from scratch.
  • Limitations of existing training: Backpropagation also requires complete knowledge of the forward computation graph, making it impossible when the forward pass contains a black-box physical component.Digital-twin or in-silico training can additionally increase training speed and energy costs and introduce a simulation–reality gap.
  • Proposed approach: The paper proposes model-free forward-forward training, a biologically plausible algorithm that directly trains arbitrary physical neural networks without modeling their nonlinear layers or building a digital twin.The approach replaces the conventional backward pass with a single forward pass through the physical system.
  • Results and validation: Compared with in-silico and backpropagation-based schemes, the method improves training speed, reduces digital computations and memory use, lowers training power consumption, and remains adaptable under unpredictable perturbations.Experiments use acoustic, microwave, and optical wave-based systems for vowel and image classification, with different wave phenomena and nonlinearities.

Model-free forward-forward training

The proposed model-free forward-forward architecture trains deep physical neural networks through local, contrastive physical computations rather than requiring detailed models of nonlinear layers. It combines nonlinear physical transformations with trainable linear multiplications and supports non-differentiable components.

  • Architecture: The architecture uses layers of nonlinear physical data transformers augmented by trainable linear multiplications.Each layer passes its output hierarchically to the next layer, while the augmented linear multiplications are trained.
  • Comparison: Unlike traditional deep reservoir computing, the architecture trains all augmented linear multiplications rather than only the final layer.The comparison concerns the trainability of the linear multiplications in the physical network.
  • Learning algorithm: The model-free forward-forward implementation trains the augmented matrices locally by minimizing a goodness-based loss.The approach applies the model-free version of the forward-forward algorithm to the physical-network architecture.
  • Learning algorithm: Training replaces backward propagation with positive and negative forward passes using correctly and incorrectly labeled inputs.The method computes a layer-wise goodness function from the squared activities of both physical passes.
  • Inference: Inference evaluates each candidate label separately and selects the label with the highest accumulated goodness across layers.A particular label is input into the physical network, and the resulting layer-wise goodness values are accumulated for classification.
  • Scope: The method can integrate non-differentiable physical systems or components between network layers.This extends the architecture beyond differentiable physical transformations.

Diverse PNNs for vowel and image classification

The paper experimentally trains diverse wave-based physical neural networks for vowel and image classification, using different physical wave phenomena and nonlinearities. Acoustic, microwave, and optical PNNs achieve high classification accuracy across these tasks.

  • Diverse physical systems: Three distinct wave-based physical systems are used, each with a different underlying wave phenomenon and source of physical nonlinearity.The systems exploit physical mixing processes, including multiple scattering, to provide linear mixing and enhance nonlinearity.
  • Acoustics-PNN: 97.62% train and 94.23% test accuracy are achieved by the two-layer acoustic-PNN on vowel classification.MF-FF performs similarly to ideal BP and outperforms in-silico training, which reaches only about 60% vowel accuracy.
  • Microwave-PNN: Around 96% accuracy on unseen test data is achieved by the three-layer microwave-PNN after roughly 20 training epochs.The microwave-PNN uses trainable matrix multiplication with physical microwave layers and skip connections.
  • Optics-PNN: The optical setup maps encoded positive and negative data through a multimode fiber and measures the resulting optical intensity on a camera.The physical system performs a complex spatiotemporal transformation involving many propagation modes.
  • Optics-PNN: 98.93% train and 96.73% test accuracy are obtained on vowels, while 99.56% train and 97.21% test accuracy are obtained on Mnist.These results come from a two-layer optics-PNN using an SLM, multimode fiber, and camera-based readout.

Real-Time Adaptable Learning

The paper evaluates MF-FF when trained deep optical PNNs encounter unpredictable physical perturbations during inference. MF-FF restores accuracy after perturbation, whereas PA-BP struggles as perturbation intensity increases.

  • Perturbation setup: A six-layer optics-PNN is perturbed during inference by adding Gaussian noise to its multimode-fiber transmission matrix.The perturbation models changes in the physical transformation function, such as altered fiber states or optical-component positions.
  • Perturbation setup: Test accuracy drops after the perturbation, motivating retraining of the optics-PNN for recovery.The perturbation is controlled using Gaussian noise with specified mean µ and standard deviation σ.
  • Comparison with PA-BP: PA-BP test accuracy oscillates around 55% under a small perturbation and worsens for more intense perturbations.PA-BP uses a digital backward-pass model while retaining the physical system for the forward pass.
  • Comparison with PA-BP: MF-FF recovers accuracy after a few epochs regardless of the applied perturbation intensity.MF-FF executes both forward passes through the physical hardware rather than relying on digital models.
  • Implication: The authors report robustness and adaptability of MF-FF in dynamic or unpredictable environments.In contrast, PA-BP’s digital model can become inaccurate after hard perturbation, requiring retraining with a new dataset.

Discussion

The discussion positions MF-FF as a hardware-compatible training approach that improves physical neural-network training while retaining robustness to perturbations. The authors also identify unresolved hardware-scaling and operation-implementation challenges that favor hybrid systems in the near term.

  • Training and efficiency: MF-FF enables forward passes through physical systems, producing a reported speed-up during both inference and training.The discussion links this approach to reducing digital computation in physical neural-network workflows.
  • Robustness and depth: MF-FF is reported to outperform both in-silico training and the state-of-the-art PA-BP method, with the gap widening as network depth increases.The authors attribute PA-BP’s increasing sensitivity to accumulated gradient mismatch across layers, whereas MF-FF trains layers separately without a model or direct dependence on other layers.
  • Broader implications: MF-FF is presented as a candidate for training optical transformer models that could achieve energy-efficiency and speed advantages.The discussion connects this opportunity to prior inference-time energy-efficiency results for optical transformer models.
  • Limitations and scope: Physical neural networks still face challenges in implementing normalization operations and scaling compactly to billions of parameters.These unresolved issues limit replacement of digital processors by physical systems.
  • Limitations and scope: The authors therefore expect near-term PNNs to support efficient hybrid in-hardware training and inference rather than fully replace digital processing units.This conclusion follows the stated implementation and scaling constraints.

Contributions

The supplied contribution section is administrative material listing authorship, supplementary sections, and paper metadata rather than scientific contributions.

  • Author contributions: The listed authors are assigned responsibilities spanning the computational engine, simulations, acoustic, optics, and microwave experiments.The text also assigns supervision, interpretation, and manuscript-writing contributions.
  • Document metadata: The document identifies the work as “Backpropagation-free Training of Deep Physical” and lists its authors.The supplied metadata also indicates that the PDF contains supplementary sections.
  • Supplementary structure: The supplementary material includes sections on methods, PNN training algorithms, acoustics-PNN, microwave-PNN, and energy-consumption analysis.Additional subsections cover input-output transformation characterization for the acoustic, microwave, and energy analyses.

Section S1. Materials and Methods

The supplementary methods describe acoustic, microwave, and optical physical systems used to realize nonlinear physical neural-network transformations. They specify the hardware configurations, input-output measurements, and sources of nonlinearity for each platform.

  • Acoustic system: The acoustic system uses a 2 m × 1 m × 0.2 m cavity with 11 propagating modes below 500 Hz, 10 loudspeakers, diffusers, and 10 nonlinear resonators.The resonators are randomly positioned on the cavity’s top layer.
  • Acoustic system: Each acoustic resonator uses microphone feedback to assign current according to i(t) = GNL|pf(t)|αNL, with GNL and αNL controlling nonlinearity.The measured front pressure pf is used in the real-time control loop.
  • Microwave system: The microwave platform uses a programmable metasurface with 76 two-bit meta-atoms covering 8% of the cavity surface over a 400 MHz interval centered on 5.2 GHz.The two control bits correspond to orthogonal field polarizations.
  • Microwave system: Microwave inputs are grouped into 40 macro-pixels, while intensities at 20 decorrelated frequencies serve as outputs.All meta-atoms within each macro-pixel share the same configuration.
  • Optical system: The optical system maps a two-dimensional input optical field to a complex output field through a multimode fiber and an experimentally obtained transmission matrix.The system uses a 532 nm, 1 mW continuous-wave source and approximately 1050 fiber modes for one polarization.

Section S2. Comparison of different PNN training algorithms

This section contrasts backpropagation-based physical-neural-network training with model-free forward-forward training. MF-FF replaces layerwise backward propagation with two forward passes and is implemented through a simple deep physical architecture with trainable linear components.

  • Comparison of training algorithms: Ideal BP, in-silico training, and physics-aware BP differ in whether the exact, modeled, or physical transformation is used during forward and backward passes.All three methods retain backpropagation as their core training procedure.
  • MF-FF: MF-FF employs two forward passes without backpropagation through layers, using contrasting objectives for positive and negative data.The positive pass increases hidden-layer goodness for real data, while the negative pass decreases it for negative data.
  • MF-FF: The method uses a balanced contrastive loss, while other imbalanced positive-negative losses are also possible.The balanced contrastive loss is identified as the loss used in this paper.
  • Motivation: Conventional forward-forward updates for fully connected networks require knowledge of the nonlinear unit, activation function, and may incur slow convergence, local minima, or heavy digital computation.These constraints motivate the paper’s simplified physical architecture.
  • Algorithm 1: The proposed MF-FF algorithm updates trainable layer matrices through forward passes and layerwise loss minimization in external and internal training loops.Algorithm 1 takes training data and epoch counts as inputs and outputs trainable parameters for each layer.

Section S3. Acoustics-PNN

The acoustics-PNN combines an actively nonlinear acoustic cavity with a two-layer MF-FF network using skip connections. Its physical measurements support surrogate modeling and comparison with alternative training procedures.

  • Experimental setup: The acoustic apparatus contains a 2 m × 1 m × 0.2 m cavity, 10 sources, 10 nonlinear meta-scatterers, and 14 static cylindrical scatterers.The cavity supports 11 propagating modes below 500 Hz.
  • Experimental setup: Training uses 4 of 10 speaker sources, each providing a 10-frequency waveform spanning 350–500 Hz, with four nonlinear output pressure waveforms.The outputs are measured at four locations of the active control system.
  • Active nonlinearity: The active acoustic control assigns resonator current from measured front pressure using tunable parameters GNL and αNL that produce system nonlinearity.The parameters are selected for high nonlinearity and control stability.
  • Acoustics-PNN: The proposed acoustics-PNN has two MF-FF-trained layers with skip connections because the acoustic transformation lacks identity-operation support and can lose information.The skip connections also allow an ensemble of subnetworks and support robustness to stochastic component failures.
  • Training comparison: The in-silico model’s inference accuracy decreases when a gap exists between the physical system and its digital forward model.The comparison introduces Gaussian parameter noise with standard deviation 0.025 to represent this mismatch.

Section S4. Microwave-PNN

The microwave-PNN uses a programmable metasurface to create structural and readout nonlinearities, with 20 frequency outputs and MF-FF training. Supplementary characterization shows frequency-dependent stirring and significant structural nonlinearity, while a digital surrogate approximates the physical forward pass.

  • Microwave hardware: The microwave cavity uses a programmable metasurface whose 76 meta-atoms cover 8% of the cavity surface and operate across a 400 MHz interval centered on 5.2 GHz.Each meta-atom has two control bits assigned to orthogonal field polarizations.
  • Microwave-PNN: The metasurface groups 152 one-bit degrees of freedom into 40 macro-pixels and uses intensities at 20 decorrelated frequencies as outputs.The input vector therefore has dimension 1 × 40, while the output vector has dimension 1 × 20.
  • Nonlinearity characterization: The input-output mapping is nonlinear because of structural nonlinearity and an additional readout nonlinearity from using intensity rather than complex-valued fields.The structural component is characterized separately from the readout contribution.
  • Supplementary characterization: Stirring efficiency is frequency dependent because it is determined by both the metasurface response and the cavity’s frequency-dependent scattering properties.The K-factor quantifies the ratio of unstirred to stirred field components, with lower values indicating more effective stirring.
  • Supplementary characterization: The linearity metric ζ is lowest near the middle of the operating interval and remains near 10 dB, evidencing significant structural nonlinearity.The metric compares transmission-spectrum variation with the error of a best-fit linear model on unseen data.

Section S5. Optics-PNN

The optics-PNN applies a three-step mode-mixing transformation and uses a two-layer architecture trained by MF-FF for vowel and MNIST classification tasks.

  • Transformation: The MMF transformation applies a Fourier transform, multiplies by a transmission matrix, and then applies an inverse Fourier transform.This transformation is illustrated for ten randomly selected MNIST digits.
  • Architecture and training: The proposed optics-PNN contains two layers and is trained using MF-FF.Matrix multiplication dimensions are 51 × 51 for the vowel dataset and 676 × 676 for MNIST.
  • Demonstration: The optical transformation is demonstrated on ten randomly selected digits from the MNIST dataset.The figure displays the transformed digits produced by the optics-PNN.

Section S6. Energy consumption and computing rate analysis

The analysis decomposes physical neural-network energy into loading, feedforward, readout, and gradient-update costs, then compares scaling and computing-rate constraints. Physical computation can scale more favorably in energy, but modulator refresh rates constrain training speed and remain an improvement target.

  • Energy components: Eload includes memory reads, DAC conversion, and wave-modulator energy, while Ereadout includes wave amplification and ADC conversion.Efeedforward covers wave energy and active system components such as SLMs or DMDs.
  • Energy analysis: Training energy is modeled as Etraining = 2Eload + Efeedforward + Ereadout + Egradient.The gradient term includes digital parameter updates and an additional physical update cost.
  • Scaling: PNN loading, readout, and feedforward costs scale linearly with data points N, whereas Egradient has an N2 scaling contribution.The analysis assumes the system has the same number of inputs and outputs.
  • Digital and physical costs: The FF-model requires quadratic N2 digital MAC operations for a forward pass, while a PNN scales linearly as N(Eload + Ereadout).The PNN’s feedforward energy is described as virtually free, with loading and readout as leading contributors.
  • Digital and physical costs: Loading and readout costs can amortize in large models, and PNNs can have a lower per-MAC cost than digital models.The cited analysis identifies the extra physical-system costs as leading contributors to total PNN energy.
  • Computing rate: PNN computing rate can be limited by wave modulators, with SLMs and DMDs operating on the order of Hz and kHz.For inference, input-data rates can reach approximately GHz, whereas training is limited by the refresh rates of modulators containing learnable weights.
  • Computing rate: Switching from an LC SLM to a DMD could improve inference speed by nearly three orders of magnitude, reaching up to 23000 frames/second.Training remains slower because its rate is limited by modulator refresh rates, although inference is faster and training occurs only once.
  • Limitations and outlook: Further improvements to modulator electronics could improve PNN training efficiency in the future.This limitation follows from the slow computation rate associated with current physical modulators.
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